Debiasing optimal transport: classical and entropic

TL;DR

This paper explores the concept of debiasability in optimal transport costs, analyzing conditions under which costs can be decomposed and extended across classical, entropic, and unbalanced regimes.

Contribution

It introduces a unified framework for debiasing optimal transport costs, extending previous results and providing new decomposition formulas for entropic optimal transport.

Findings

Identifies conditions for debiasability in various optimal transport regimes.

Provides new decomposition formulas for entropic optimal transport.

Extends results to unbalanced optimal transport settings.

Abstract

We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}\psi(x,z)+\psi(y,z)$ for some set $\mathscr{Z}$ and some function $\psi: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings and we generalize in this way the findings of \cite{feydy2019interpolating} and \cite{sejourne2019sinkhorn}. As a byproduct, we derive novel decomposition formulas for entropic optimal transport, which may be of independent interest.